Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B

Percentage Accurate: 90.5% → 95.2%

Time: 14.6s

Alternatives: 7

Speedup: 0.5×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+264}:\\ \;\;\;\;x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \left(z \cdot \left(y \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 5e+264)
   (- (* x x) (+ (* -4.0 (* t y)) (* 4.0 (* z (* y z)))))
   (pow x 2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5e+264) {
		tmp = (x * x) - ((-4.0 * (t * y)) + (4.0 * (z * (y * z))));
	} else {
		tmp = pow(x, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 5d+264) then
        tmp = (x * x) - (((-4.0d0) * (t * y)) + (4.0d0 * (z * (y * z))))
    else
        tmp = x ** 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5e+264) {
		tmp = (x * x) - ((-4.0 * (t * y)) + (4.0 * (z * (y * z))));
	} else {
		tmp = Math.pow(x, 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 5e+264:
		tmp = (x * x) - ((-4.0 * (t * y)) + (4.0 * (z * (y * z))))
	else:
		tmp = math.pow(x, 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 5e+264)
		tmp = Float64(Float64(x * x) - Float64(Float64(-4.0 * Float64(t * y)) + Float64(4.0 * Float64(z * Float64(y * z)))));
	else
		tmp = x ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 5e+264)
		tmp = (x * x) - ((-4.0 * (t * y)) + (4.0 * (z * (y * z))));
	else
		tmp = x ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+264], N[(N[(x * x), $MachinePrecision] - N[(N[(-4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000033e264

   1. Initial program 96.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around 0 95.1%

      \[\leadsto x \cdot x - \color{blue}{\left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \left(y \cdot {z}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. add-cube-cbrt 94.9%

         \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \color{blue}{\left(\left(\sqrt[3]{y \cdot {z}^{2}} \cdot \sqrt[3]{y \cdot {z}^{2}}\right) \cdot \sqrt[3]{y \cdot {z}^{2}}\right)}\right) \]

      2. pow3 94.9%

         \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \color{blue}{{\left(\sqrt[3]{y \cdot {z}^{2}}\right)}^{3}}\right) \]

   4. Applied egg-rr 94.9%

      \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \color{blue}{{\left(\sqrt[3]{y \cdot {z}^{2}}\right)}^{3}}\right) \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 95.1%

         \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}\right) \]

      2. unpow2 95.1%

         \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      3. associate-*r* 98.9%

         \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

   6. Applied egg-rr 98.9%

      \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

   if 5.00000000000000033e264 < (*.f64 x x)

   1. Initial program 77.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in x around inf 91.9%

      \[\leadsto \color{blue}{{x}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+264}:\\ \;\;\;\;x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \left(z \cdot \left(y \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \]

Alternative 2: 93.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right) \leq \infty:\\ \;\;\;\;x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \left(z \cdot \left(y \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (* x x) (* (* y 4.0) (- t (* z z)))) INFINITY)
   (- (* x x) (+ (* -4.0 (* t y)) (* 4.0 (* z (* y z)))))
   (* t (* y 4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) + ((y * 4.0) * (t - (z * z)))) <= ((double) INFINITY)) {
		tmp = (x * x) - ((-4.0 * (t * y)) + (4.0 * (z * (y * z))));
	} else {
		tmp = t * (y * 4.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) + ((y * 4.0) * (t - (z * z)))) <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) - ((-4.0 * (t * y)) + (4.0 * (z * (y * z))));
	} else {
		tmp = t * (y * 4.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) + ((y * 4.0) * (t - (z * z)))) <= math.inf:
		tmp = (x * x) - ((-4.0 * (t * y)) + (4.0 * (z * (y * z))))
	else:
		tmp = t * (y * 4.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z)))) <= Inf)
		tmp = Float64(Float64(x * x) - Float64(Float64(-4.0 * Float64(t * y)) + Float64(4.0 * Float64(z * Float64(y * z)))));
	else
		tmp = Float64(t * Float64(y * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) + ((y * 4.0) * (t - (z * z)))) <= Inf)
		tmp = (x * x) - ((-4.0 * (t * y)) + (4.0 * (z * (y * z))));
	else
		tmp = t * (y * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x * x), $MachinePrecision] - N[(N[(-4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t))) < +inf.0

   1. Initial program 96.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around 0 95.6%

      \[\leadsto x \cdot x - \color{blue}{\left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \left(y \cdot {z}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. add-cube-cbrt 95.5%

         \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \color{blue}{\left(\left(\sqrt[3]{y \cdot {z}^{2}} \cdot \sqrt[3]{y \cdot {z}^{2}}\right) \cdot \sqrt[3]{y \cdot {z}^{2}}\right)}\right) \]

      2. pow3 95.5%

         \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \color{blue}{{\left(\sqrt[3]{y \cdot {z}^{2}}\right)}^{3}}\right) \]

   4. Applied egg-rr 95.5%

      \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \color{blue}{{\left(\sqrt[3]{y \cdot {z}^{2}}\right)}^{3}}\right) \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 95.6%

         \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}\right) \]

      2. unpow2 95.6%

         \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      3. associate-*r* 99.1%

         \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

   6. Applied egg-rr 99.1%

      \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t)))

   1. Initial program 0.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around 0 23.1%

      \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(t \cdot y\right)} \]
   3. Step-by-step derivation

      1. *-commutative 23.1%

         \[\leadsto x \cdot x - -4 \cdot \color{blue}{\left(y \cdot t\right)} \]

   4. Simplified 23.1%

      \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(y \cdot t\right)} \]

   5. Taylor expanded in x around 0 31.8%

      \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
   6. Step-by-step derivation

      1. *-commutative 31.8%

         \[\leadsto 4 \cdot \color{blue}{\left(y \cdot t\right)} \]

      2. associate-*r* 39.1%

         \[\leadsto \color{blue}{\left(4 \cdot y\right) \cdot t} \]

      3. *-commutative 39.1%

         \[\leadsto \color{blue}{\left(y \cdot 4\right)} \cdot t \]

   7. Simplified 39.1%

      \[\leadsto \color{blue}{\left(y \cdot 4\right) \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right) \leq \infty:\\ \;\;\;\;x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \left(z \cdot \left(y \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \end{array} \]

Alternative 3: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* x x) (* (* y 4.0) (- t (* z z))))))
   (if (<= t_1 INFINITY) t_1 (* t (* y 4.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t * (y * 4.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t * (y * 4.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) + ((y * 4.0) * (t - (z * z)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = t * (y * 4.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(t * Float64(y * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = t * (y * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t))) < +inf.0

   1. Initial program 96.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t)))

   1. Initial program 0.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around 0 23.1%

      \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(t \cdot y\right)} \]
   3. Step-by-step derivation

      1. *-commutative 23.1%

         \[\leadsto x \cdot x - -4 \cdot \color{blue}{\left(y \cdot t\right)} \]

   4. Simplified 23.1%

      \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(y \cdot t\right)} \]

   5. Taylor expanded in x around 0 31.8%

      \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
   6. Step-by-step derivation

      1. *-commutative 31.8%

         \[\leadsto 4 \cdot \color{blue}{\left(y \cdot t\right)} \]

      2. associate-*r* 39.1%

         \[\leadsto \color{blue}{\left(4 \cdot y\right) \cdot t} \]

      3. *-commutative 39.1%

         \[\leadsto \color{blue}{\left(y \cdot 4\right)} \cdot t \]

   7. Simplified 39.1%

      \[\leadsto \color{blue}{\left(y \cdot 4\right) \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right) \leq \infty:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \end{array} \]

Alternative 4: 66.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ x \cdot x - -4 \cdot \left(t \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* -4.0 (* t y))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - (-4.0 * (t * y));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((-4.0d0) * (t * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - (-4.0 * (t * y));
}

Python

def code(x, y, z, t):
	return (x * x) - (-4.0 * (t * y))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(-4.0 * Float64(t * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - (-4.0 * (t * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

2. Taylor expanded in z around 0 71.1%

   \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(t \cdot y\right)} \]
3. Step-by-step derivation

   1. *-commutative 71.1%

      \[\leadsto x \cdot x - -4 \cdot \color{blue}{\left(y \cdot t\right)} \]

4. Simplified 71.1%

   \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(y \cdot t\right)} \]

5. Final simplification 71.1%

   \[\leadsto x \cdot x - -4 \cdot \left(t \cdot y\right) \]

Alternative 5: 6.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \left(t \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* -4.0 (* t y)))

C

double code(double x, double y, double z, double t) {
	return -4.0 * (t * y);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-4.0d0) * (t * y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return -4.0 * (t * y);
}

Python

def code(x, y, z, t):
	return -4.0 * (t * y)

Julia

function code(x, y, z, t)
	return Float64(-4.0 * Float64(t * y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -4.0 * (t * y);
end

Wolfram

code[x_, y_, z_, t_] := N[(-4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

2. Taylor expanded in t around inf 33.2%

   \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
3. Step-by-step derivation

   1. *-commutative 33.2%

      \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot 4} \]

   2. *-commutative 33.2%

      \[\leadsto \color{blue}{\left(y \cdot t\right)} \cdot 4 \]

4. Simplified 33.2%

   \[\leadsto \color{blue}{\left(y \cdot t\right) \cdot 4} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 18.8%

      \[\leadsto \color{blue}{\sqrt{\left(y \cdot t\right) \cdot 4} \cdot \sqrt{\left(y \cdot t\right) \cdot 4}} \]

   2. sqrt-unprod 22.6%

      \[\leadsto \color{blue}{\sqrt{\left(\left(y \cdot t\right) \cdot 4\right) \cdot \left(\left(y \cdot t\right) \cdot 4\right)}} \]

   3. *-commutative 22.6%

      \[\leadsto \sqrt{\color{blue}{\left(4 \cdot \left(y \cdot t\right)\right)} \cdot \left(\left(y \cdot t\right) \cdot 4\right)} \]

   4. *-commutative 22.6%

      \[\leadsto \sqrt{\left(4 \cdot \left(y \cdot t\right)\right) \cdot \color{blue}{\left(4 \cdot \left(y \cdot t\right)\right)}} \]

   5. swap-sqr 22.6%

      \[\leadsto \sqrt{\color{blue}{\left(4 \cdot 4\right) \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right)}} \]

   6. metadata-eval 22.6%

      \[\leadsto \sqrt{\color{blue}{16} \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right)} \]

   7. metadata-eval 22.6%

      \[\leadsto \sqrt{\color{blue}{\left(-4 \cdot -4\right)} \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right)} \]

   8. swap-sqr 22.6%

      \[\leadsto \sqrt{\color{blue}{\left(-4 \cdot \left(y \cdot t\right)\right) \cdot \left(-4 \cdot \left(y \cdot t\right)\right)}} \]

   9. sqrt-unprod 6.0%

      \[\leadsto \color{blue}{\sqrt{-4 \cdot \left(y \cdot t\right)} \cdot \sqrt{-4 \cdot \left(y \cdot t\right)}} \]

   10. add-sqr-sqrt 6.7%

       \[\leadsto \color{blue}{-4 \cdot \left(y \cdot t\right)} \]

   11. expm1-log1p-u 6.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-4 \cdot \left(y \cdot t\right)\right)\right)} \]

   12. expm1-udef 6.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-4 \cdot \left(y \cdot t\right)\right)} - 1} \]

   13. *-commutative 6.6%

       \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \color{blue}{\left(t \cdot y\right)}\right)} - 1 \]

   14. associate-*r* 6.6%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-4 \cdot t\right) \cdot y}\right)} - 1 \]

   15. *-commutative 6.6%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left(-4 \cdot t\right)}\right)} - 1 \]

6. Applied egg-rr 6.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y \cdot \left(-4 \cdot t\right)\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 6.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(-4 \cdot t\right)\right)\right)} \]

   2. expm1-log1p 6.7%

      \[\leadsto \color{blue}{y \cdot \left(-4 \cdot t\right)} \]

   3. *-commutative 6.7%

      \[\leadsto \color{blue}{\left(-4 \cdot t\right) \cdot y} \]

   4. associate-*r* 6.7%

      \[\leadsto \color{blue}{-4 \cdot \left(t \cdot y\right)} \]

8. Simplified 6.7%

   \[\leadsto \color{blue}{-4 \cdot \left(t \cdot y\right)} \]

9. Final simplification 6.7%

   \[\leadsto -4 \cdot \left(t \cdot y\right) \]

Alternative 6: 31.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(t \cdot 4\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* y (* t 4.0)))

C

double code(double x, double y, double z, double t) {
	return y * (t * 4.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y * (t * 4.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return y * (t * 4.0);
}

Python

def code(x, y, z, t):
	return y * (t * 4.0)

Julia

function code(x, y, z, t)
	return Float64(y * Float64(t * 4.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y * (t * 4.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

2. Taylor expanded in t around inf 33.2%

   \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
3. Step-by-step derivation

   1. associate-*r* 33.2%

      \[\leadsto \color{blue}{\left(4 \cdot t\right) \cdot y} \]

   2. *-commutative 33.2%

      \[\leadsto \color{blue}{y \cdot \left(4 \cdot t\right)} \]

   3. *-commutative 33.2%

      \[\leadsto y \cdot \color{blue}{\left(t \cdot 4\right)} \]

4. Simplified 33.2%

   \[\leadsto \color{blue}{y \cdot \left(t \cdot 4\right)} \]

5. Final simplification 33.2%

   \[\leadsto y \cdot \left(t \cdot 4\right) \]

Alternative 7: 31.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ t \cdot \left(y \cdot 4\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* t (* y 4.0)))

C

double code(double x, double y, double z, double t) {
	return t * (y * 4.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = t * (y * 4.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return t * (y * 4.0);
}

Python

def code(x, y, z, t):
	return t * (y * 4.0)

Julia

function code(x, y, z, t)
	return Float64(t * Float64(y * 4.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = t * (y * 4.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

2. Taylor expanded in z around 0 71.1%

   \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(t \cdot y\right)} \]
3. Step-by-step derivation

   1. *-commutative 71.1%

      \[\leadsto x \cdot x - -4 \cdot \color{blue}{\left(y \cdot t\right)} \]

4. Simplified 71.1%

   \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(y \cdot t\right)} \]

5. Taylor expanded in x around 0 33.2%

   \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
6. Step-by-step derivation

   1. *-commutative 33.2%

      \[\leadsto 4 \cdot \color{blue}{\left(y \cdot t\right)} \]

   2. associate-*r* 33.6%

      \[\leadsto \color{blue}{\left(4 \cdot y\right) \cdot t} \]

   3. *-commutative 33.6%

      \[\leadsto \color{blue}{\left(y \cdot 4\right)} \cdot t \]

7. Simplified 33.6%

   \[\leadsto \color{blue}{\left(y \cdot 4\right) \cdot t} \]

8. Final simplification 33.6%

   \[\leadsto t \cdot \left(y \cdot 4\right) \]

Developer target: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* 4.0 (* y (- (* z z) t)))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((z * z) - t)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - (4.0d0 * (y * ((z * z) - t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((z * z) - t)));
}

Python

def code(x, y, z, t):
	return (x * x) - (4.0 * (y * ((z * z) - t)))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(4.0 * Float64(y * Float64(Float64(z * z) - t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - (4.0 * (y * ((z * z) - t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(4.0 * N[(y * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023326 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))